⊥n as an Aec

Definition 1.1. A class of τ -structures, (K,≺K), is said to be an abstract elementary class (AEC) if both K and the binary relation ≺K are closed under isomorphism and satisfy the following conditions. • A1. IfM ≺K N thenM ⊆ N . • A2. ≺K is a partial order on K. • A3. If 〈Ai : i < δ〉 is continuous ≺K -increasing chain: (1) ⋃ i<δ Ai ∈K; (2) for each j < δ, Aj ≺K ⋃ i<δ Ai (3) if each Ai ≺K M ∈K then ⋃ i<δ Ai ≺K M . • A4. If A,B,C ∈K, A ≺K C, B ≺K C and A ⊆ B then A ≺K B. • A5. There is a Löwenheim-Skolem number LS(K) such that if A ⊆ B ∈ K there is a A′ ∈K with A ⊆ A′ ≺K B and |A ′| ≤ |A|+ κ(K). We use ‘module’ to mean a right R-module; we will work with various properties of R.